A linear uncertain pharmacokinetic model driven by Liu process. (English) Zbl 1481.92053
Summary: There are inevitably dynamic noises in the pharmacokinetics, which can not be depicted well by deterministic methods. In addition, stochastic pharmacokinetic models under the framework of probability theory are valid under the premise that the distribution function (no matter how we get it) is close enough to the real frequency. However, due to the complexity and changeability of the world, economic and technological reasons, this premise can not be satisfied in some cases. Under these situations, this paper first deduces a linear uncertain pharmacokinetic model based on uncertain differential equations to rationally model dynamic noises. Several essential pharmacokinetic indexes are investigated. Generalized moment estimations for unknown parameters in this linear uncertain pharmacokinetic model are also given. A numerical example and two case studies using real data illustrate our method and compare the linear uncertain pharmacokinetic model driven by Liu process with deterministic pharmacokinetic model. As we can see, the linear uncertain pharmacokinetic model driven by Liu processes gives satisfactory results, and is suitable to model pharmacokinetics in reality.
MSC:
| 92C45 | Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.) |
| 60H30 | Applications of stochastic analysis (to PDEs, etc.) |
Keywords:
uncertain differential equations; Liu process; pharmacokinetics; linear uncertain pharmacokinetic model; intravenous infusionReferences:
| [1] | Gibaldi, M.; Perrier, D., Pharmacokinetics (2007), Informa Healthcare: Informa Healthcare USA |
| [2] | Hasegawa, C.; Duffull, S., Exploring inductive linearization for pharmacokinetic-pharmacodynamic systems of nonlinear ordinary differential equations, J. Pharmacokinet. Pharm., 45, 1, 35-47 (2018) |
| [3] | Article 119 |
| [4] | Ramanathan, M., An application of ITO’s lemma in population pharmacokinetics and pharmacodynamics, Pharm. Res., 16, 4, 584-586 (1999) |
| [5] | Ditlevsen, S.; Gaetano, A., Stochastic vs. deterministic uptake of dodecanedioic acid by isolated rat livers, Br. Math. Biol., 67, 3, 547-561 (2005) · Zbl 1334.92133 |
| [6] | Liu, B., Uncertainty Theory (2007), Springer: Springer Berlin · Zbl 1141.28001 |
| [7] | Article 1 |
| [8] | Ferrante, L.; Bompader, S.; Leone, L., A stochastic compartmental model with long lasting infusion, Biometrical J., 45, 2, 182-194 (2013) · Zbl 1441.62345 |
| [9] | Sen, P.; Bell, D., A model for the interaction of two chemicals, J. Theor. Biol., 238, 3, 652-656 (2006) · Zbl 1445.92133 |
| [10] | Liu, B., Some research problems in uncertainty theory, J. Uncertain. Syst., 3, 1, 3-10 (2009) |
| [11] | Liu, B., Fuzzy process, hybrid process and uncertain process, J. Uncertain. Syst., 2, 1, 3-16 (2008) |
| [12] | Chen, X.; Liu, B., Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optim. Decis. Ma., 9, 1, 69-81 (2010) · Zbl 1196.34005 |
| [13] | Yao, K.; Gao, J.; Gao, Y., Some stability theorems of uncertain differential equation, Fuzzy Optim. Decis. Ma., 12, 1, 3-13 (2013) · Zbl 1412.60104 |
| [14] | Yao, K.; Ke, H.; Sheng, Y., Stability in mean for uncertain differential equation, Fuzzy Optim. Decis. Ma., 14, 3, 365-379 (2015) · Zbl 1463.93250 |
| [15] | Article 8 |
| [16] | Liu, Y., Analytic method for solving uncertain differential equations, J. Uncertain. Syst., 6, 4, 243-248 (2012) |
| [17] | Yao, K.; Chen, X., A numerical method for solving uncertain differential equations, J. Intell. Fuzzy Syst., 25, 3, 825-832 (2013) · Zbl 1291.65025 |
| [18] | Article 17 |
| [19] | Yang, X.; Ralescu, D., Adams method for solving uncertain differential equations, Appl. Math. Comput., 270, 993-1003 (2015) · Zbl 1410.65012 |
| [20] | Yao, K., Uncertain Differential Equations (2016), Springer-Verlag: Springer-Verlag Berlin · Zbl 1378.60009 |
| [21] | Yao, K.; Liu, B., Parameter estimation in uncertain differential equations, Fuzzy Optim. Decis. Ma., 19, 1, 1-12 (2020) · Zbl 1437.60033 |
| [22] | Yang, X.; Liu, Y.; Park, G. K., Parameter estimation of uncertain differential equation with application to financial market, Chaos Soliton Fract., 139, 110026 (2020) · Zbl 1490.91201 |
| [23] | Wang, L.; Xiong, C.; Wang, X., A dimension-wise method and its improvement for multidisciplinary interval uncertainty analysis, Appl. Math. Model., 59, 680-695 (2018) · Zbl 1480.62250 |
| [24] | Wang, L.; Liu, Y.; Liu, Y., An inverse method for distributed dynamic load identification of structures with interval uncertainties, Adv. Eng. Softw., 131, 77-89 (2019) |
| [25] | Xiong, C.; Wang, L.; Liu, G.; Shi, Q., An iterative dimension-by-dimension method for structural interval response prediction with multidimensional uncertain variables, Aerosp. Sci. Technol., 86, 572-581 (2019) |
| [26] | Li, Z.; Sheng, Y.; Teng, Z.; Miao, H., An uncertain differential equation for SIS epidemic model, J. Intell. Fuzzy Syst., 33, 2317-2327 (2017) · Zbl 1377.92094 |
| [27] | Zhang, Z.; Yang, X., Uncertain population model, Soft Comput., 24, 2417-2423 (2020) · Zbl 1436.92012 |
| [28] | Dipiro, J.; Talbert, R.; Yee, G., Pharmacotherapy: A Pathophysiologic Approach (2011), McGraw-Hill Medical: McGraw-Hill Medical USA |
| [29] | Liu, B., Uncertainty Theory (2015), Springer Berlin · Zbl 1309.28001 |
| [30] | Metzger, G.; Massari, C.; Etienne, M., Spontaneous or imposed circadian changes in plasma concentrations of 5-fluorouracil coadministered with folinic acid and oxaliplatin: Relationship with mucosal toxicity in patients with cancer, Clin. Pharmacol. Ther., 56, 190-201 (1994) |
| [31] | Article 13279 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
